A Non-Abelian Conjecture of Tate–Shafarevich Type for Hyperbolic

Home , Jennifer Balakrishnan, Minhyong Kim, Yuri Manin

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j jp n / 1 Sel ( ) Hf (Gp,Un) X locp and we deﬁne 1 n (Z ) := j− loc (Sel ( )) . X p n p p X These form a nested sequence of subsets like so. (Z ) (Z ) (Z ) (Z). X p ⊃X p 1 ⊃X p 2 ⊃···⊃X The conjecture, which was ﬁrst proposed by M.K. in his lectures at the I.H.E.S. in February of 2012, is as follows. ANON-ABELIANCONJECTURE 3

Conjecture (3.1 below). Equality (Z ) = (Z) is obtained for large n. X p n X n 1.4. We also suggest a variant of the Selmer scheme SelS( ) of , suited 1 X X to computing the Z[S− ]-valued points of for S a ﬁnite set of primes, by dropping the local geometricity condition overX S. This gives rise to a square 1 X(Z[S− ]) / X(Zp)

jS jp n / 1 SelS( ) Hf (Gp,Un), X locp and to an associated sequence of subsets (Z ) (Z ) (Z ) (Z), X p ⊃X p S,1 ⊃X p S,2 ⊃···⊃X for which equality (Z ) = (Z[S 1]) may hold for large n. X p S,n X − 1.5. These constructions are based on the third author’s approach to integral points, introduced in [Kim2] and [Kim3]. The relationship to the section conjecture has been explored before. For instance in [Kim6], the third author shows that if (Zp)n = (Zp) for some n, then the section conjecture would in principleX allow one6 X to obtain a computable bound on the height of rational points. Our present conjecture, however, is quite different in fla- vor from the section conjecture and its direct consequences. It shares more with the conjectures of Tate–Shafarevich and Birch–Swinnerton-Dyer, both in terms of concreteness and in terms of computability. Indeed, like the BSD conjecture, the present conjecture can actually be tested numerically. 1.6. Our principal goal below is to do just that. Work completed elsewhere allows us to verify our conjecture in a range of cases. New in this article is the case of a punctured elliptic curve of rank zero: we are able to compute the sets (Zp)2, and subsequently to verify the conjecture for many such curves. ThisX computation is based on a study of the unipotent Kummer map j : X(F ) H1(G ,U ) v v → v 2 for a punctured elliptic curve X over a local field Fv of residue characteristic v = p. Our main theorem (4.1.6) says that the p-adic height function can be6 retrieved from this map. This is of interest in its own right, and is suggestive of the possibility of obtaining functions on the local points from higher quotients of the unipotent fundamental group which might play a role similar to the role played by heights here. This point of view is implicit in the third author’s work on nonabelian reciprocity laws [Kim1] and in ongoing joint work between him and Jonathan Pridham. 4 BALAKRISHNAN,DAN-COHEN,KIM,ANDWEWERS

1.7. As explained in [Kim2, Kim3], the key to computing the map jp is its equivalence with a certain p-adic analog α : (Z ) U DR/F 0 X p → n of the higher Albanese map of Richard Hain [Hai] through a lifting of the Bloch-Kato exponential to the unipotent level obtained via the unipotent p-adic Hodge theory of Martin Olsson [Ols]. The p-adic unipotent Albanese map α is given in coordinates by certain p-adic iterated integrals (known also as Coleman functions), and there are fairly well-established methods for producing explicit formulas for the resulting iterated integrals on the one hand, and for computing p-adic approximations of their values on the other. For instance, the case of the thrice punctured line was treated by Furusho [Fur1, Fur2] and by Besser–de Jeu [BdJ]. The problem of explicit determi- nation of the unipotent Albanese map for punctured elliptic curves in depth two is treated by Kim [Kim4] and Balakrishnan–Kedlaya–Kim [BKK]. The problem of computing Coleman functions on hyperelliptic curves is treated by Balakrishnan–Bradshaw–Kedlaya [BBK] and by Balakrishnan [Bal2]. 1.8. We turn to the map loc : Seln( ) H1(G ,U ). p X → f p n This is actually an algebraic map of finite-type affine Qp-schemes; its target is in fact isomorphic to affine space. Let (n) denote the ideal defining its scheme-theoretic image. As explained inL [Kim3], as soon as (n) = 0, L 6 (Zp)n becomes finite. Moreover, several well known motivic conjectures X(Fontaine-Mazur-Jannsen, Bloch-Kato) imply that for n large,

j∗ (n + 1) ) j∗ (n), p L p L and, in fact, the larger ideal contains elements that are algebraically inde- 1 pendent of the elements in j∗ (n). So a point in the common zero set for all n should be there for a goodL reason; our conjecture expresses the belief that such a point must belong to (Z). X 1.9. The study of the ideals (n) relates not only the the plausibility of our conjecture, but also to its usefulness.L Explicit computation of these ideals has been achieved in a range of special cases. An approach to the case of the thrice punctured line using the methods of mixed Tate motives is cur- rently under development in a sequence of articles by Dan-Cohen and Wewers [DCW2, DCW3, DC]. The case of punctured elliptic curves in depth two was treated by Kim [Kim4] and Balakrishnan–Kedlaya–Kim [BKK]. The case of punctured hyperelliptic curves of genus equal to the Mordell-Weil rank of their Jacobian is treated in Balakrishnan–Besser–Müller [BBM]. The case of punctured elliptic curves of rank zero is treated in section 5 below. We

1 ∗ Technically speaking, the pullback jp appearing here may be thought of as a pullback of locally analytic functions on associated p-adic analytic spaces. ANON-ABELIANCONJECTURE 5 believe strongly in the feasibility of computing the ideals (n) and subse- L quently the loci (Zp)n (as well as their S-integral variants) in a range of cases far beyondX those mentioned above and detailed below. Such computations will provide powerful tools for bounding the number of (S-)integral points. If the conjecture holds, then bounds obtained in this way can be made sharp. 1.10. We begin in section 2 by giving a careful construction of Seln( ). Our construction, which is a bit more elaborate than indicated above,X relies on the work done in [Kim3] to endow Seln( ) with the structure of an affine, X finite-type Qp-scheme for which the map locp is algebraic. In section 3, after restating the conjecture, we discuss again in more detail its relationship to the finiteness of Ш and to the section conjecture, as well as the computability of the local Kummer map jp via the p-adic unipotent Albanese map. 1.11. The remainder of the article is devoted to discussing several special cases in which we are able to compute the loci (Zp)n,S and so to obtain numerical evidence for the conjecture. SectionX 4 is devoted to proving a preliminary theorem to be used in our study of punctured elliptic curves of rank zero in section 5 below. Fix a prime p = 2. Let F be a finite unramified extension of Q for l = p 6 v l 6 and let Ev be an elliptic curve over Fv. We let Gv denote the total Galois group of Fv. We fix a certain tangent vector b at O which serves as base point for the level 2 quotient of the p-adic étale unipotent fundamental group U of X = E O . Let 2 \ { } log χ : Gab Q v → p denote the p-adic logarithm of the cyclotomic character. Let jv denote the local unipotent Kummer map X(F ) H1(G ,U ). v → v 2 As we explain in segment 4.1.4, the map H1(G , Q (1)) H1(G ,U ) v p → v 2 induced by the inclusion Qp(1) U2 is bijective. This allows us to regard jv 1 ⊂ as a map to H (Gv, Qp(1)). Using the cup product H1(G , Q ) H1(G , Q (1)) H2(G , Q (1)) v p × v p → v p and the Hasse invariant H2(G , Q (1)) ∼ Q v p −→ p we define φ : X(F ) Q v v → p by φ (a) = log χ j (a). v ∪ v We define a p-adic local Néron function to be a function X(F ) Q v → p 6 BALAKRISHNAN,DAN-COHEN,KIM,ANDWEWERS which satisfies axioms analogous to those which define the real Néron function (see segment 4.1.6 below). Our main goal in section 4 is Theorem 4.1.6:

Theorem. The function φv is a p-adic local Néron function. Consider Weierstrass coordinates x,y in which X is given by 2 3 2 y + a1xy + a3y = x + a2x + a4x + a6. Among the three axioms which define a Néron function, verification of the formula φ (2a) = 4φ (a) log (2y + a x + a )(a) v v − | 1 3 |v is hardest. This is accomplished via an elaborate computation which takes place on the profinite level and which culminates in the theorem of segment 4.3.7. 1.12. Let = O, where is the regular minimal model of an elliptic curve withX semi-stableE \ reductionE everywhere, let α be the global 1-form given by dx α = 2y + a1x + a3 in Weierstrass coordinates, and let β be the meromorphic form β = xα. Let b be the integral tangent vector at O dual to α(O). Let S denote the set of primes of bad reduction for and for each l S, let Nl = ordl(∆ ), where ∆ is the minimal discriminant.E Define a set∈ E E W := (n(N n)/2N ) log l 0 n < N , l l − l ≤ l and for each w = (wl)l S W := l S Wl, define ∈ ∈ Q ∈ w = w . k k l Xl S ∈ Our main result in section 5 is as follows.

Theorem. Suppose has rank zero and that ШE[p∞] < , and let p be an odd prime of goodE reduction. With assumptions as above∞ (Z ) = Ψ(w), X p 2 w[W ∈ where Ψ(w) := z (Z ) log(z) = 0, D (z)= w . ∈X p 2 k k Here, z log(z)= α Zb and z D2(z)= αβ, Zb are Coleman (iterated) integral functions on (Q ). X p ANON-ABELIANCONJECTURE 7

1.13. Let us sketch the proof of theorem 1.12. It follows from theorem 5.2 of Silverman [Sil1] that the local height at primes v = p takes values in the finite set W ; by theorem 4.1.6, this applies to the image6 of (Z ) under v X v φ (Z ) H1(G , Q (1)) v Q X v → v p −→ p where φv now denotes the map c log χ c. 7→ ∪ As we explain in segment 5.3, the map locp at level 2 factors as ;( Sel2( ) H1(G , Q (1)) ֒ H1(G ,U X → f p p → f p 2 it is here that we use the assumption about the rank. Drawing on global reciprocity, we find that the image of Sel2( ) in H1(G , Q (1)) is given by X f p p η φ (η)= w for some w W . p k k ∈ As we explain in remark 5.6, this hints at the possibility of a certain non- abelian reciprocity law, an idea carried further by the third author in [Kim1]. This also translates into a proof of the theorem, through the unipotent Bloch- Kato exponential. Armed with theorem 1.12 we are able to verify conjecture 3.1 for the prime p = 5 for 256 semi-stable elliptic curves of rank zero from Cremona’s table. We also extend our discussion of punctured elliptic curves with a brief treatment of the rank-one case. 1.14. In section 6 we consider the thrice punctured line over Z. Of course, in this case the set of Z-points is empty, so the conjecture holds at level n when (Z ) = . Our results may be summarized as follows. X p n ∅ Proposition. Let = P1 0, 1, . Then X \ { ∞} (Z ) = φ X p 1 if p 2 mod3. If p 1 mod3, then ≡ ≡ (Z ) = φ X p 2 if the value of the p-adic dilogarithm Li2(z) at a sixth root of 1 is non-zero. We also report on computations showing that indeed Li (ζ ) = 0 in the range 2 6 6 3 p 105. ≤ ≤ 1.15. In section 7 we discuss curves of genus 2. We consider as an example ≥ the Fermat curve Xl given by xl + yl = zl. We find that if the Tate-Shafarevich group of the Jacobian fulfills its conjectured finiteness, if l = 5 or 7, and if p 1 mod l, then conjecture 3.1 holds at level 1. We also show how, starting6≡ with a punctured elliptic curve which fulfills conjecture 3.1 at level 2 we can construct a curve of higher genus which fulfills the conjecture at level 2 as well. 8 BALAKRISHNAN,DAN-COHEN,KIM,ANDWEWERS

1.16. Finally, in section 8 we turn to the S-integral variant of our conjecture mentioned above. We apply this to the thrice punctured line, concluding that here the conjecture holds for S = 2 and p = 3, 5, 6 in depth 2. { } Acknowledgements. M.K. is grateful to John Coates, Henri Darmon, Kazuya Kato, Florian Pop, and Andrew Wiles for a continuous stream of discussions on the topic of this paper. He is also grateful to Shinichi Mochizuki whose question prompted a precise formulation of the conjecture, and to Yuichiro Hoshi for a kind and detailed reply to a question about a pro-p analogue. We would like to thank the referee for many helpful comments. 2. Selmer schemes with stringent local conditions 2.1. We let Spec Z denote a regular minimal Z-model of a hyperbolic curve over QX. By → this we mean one of the following. – = P1 where is a reduced horizontal divisor with at least X \ D D 1 three C-points. In this case we let ′ = P . – The regular minimal model of a compactX smooth curve of genus 2. ≥ We let ′ = . – The complementX X of a non-empty reduced horizontal divisor inside a regular minimal model of a compact smooth curve of genusD 1. X ′ ≥ We fix a “base-point” b of . In all three cases b may be a Z-valued point. X In the first and third cases, suppose that with ′ open and D⊂Y1 Y ⊂X Spec Z smooth, so that in particular, Ω ′ is invertible. Then Y → / Spec Z|D we allow b to be an “integral tangent vector”,X by which we mean a nowhere vanishing section of the tangent sheaf 1 ′/ Spec Z =Ω ∨′/ Spec Z TX |D X |D to along . X ′ D 2.2. Let p denote an odd prime of good reduction. We then have the unipotent p-adic étale fundamental group U of Q at b constructed by Deligne [Del]. We denote its descending central seriesX by U = U 1 U 2 , and the associated quotients by U = U/U n+1. We also have, for⊃ every⊃x ··· (Z), n ∈X the path torsor P (x), and corresponding quotients Pn(x). We let T denote a finite set of primes which contains all primes of bad reduction for ′ and for , plus the auxiliary prime p. Let Q denote the extension of QXwhich D T is maximal for the property of being unramified outside of T , and let GT denote the Galois group of QT over Q. Then as explained in §2 of Selmer varieties [Kim3], Un possesses a GT -action, and Pn(x) bears the structure of a GT -equivariant Un-torsor, with GT acting as usual on the left, but Un acting on the right. ANON-ABELIANCONJECTURE 9

2.3. For each prime v, we ﬁx an embedding Q Q in the algebraic closure T ⊂ v of Qv. This gives us for every v a map G G v → T from the total Galois group of Q (which, for v / T , factors through Zˆ). This v ∈ also gives us an isomorphism of Un with the unipotent fundamental group of

Qv , which we continue to denote by the same symbol. For y (Zv), we Xhave the local path torsor P (y), a G -equivariant U -torsor. For∈y X (Z ), n v n ∈X p the associated torsor Pn(y) is moreover crystalline in the sense of §2 of Selmer varieties; as explained there, this follows from Olsson [Ols]. 1 2.4. For each prime v there is an affine, finite type Qp-scheme H (Gv,Un) parametrizing Gv-equivariant Un-torsors. For v = p there’s a closed subscheme H1(G ,U ) H1(G ,U ) f p n ⊂ p n which parametrizes those torsors which are crystalline. There is also the 1 global H (GT ,Un), an affine finite type Qp-scheme parametrizing GT -equivariant Un-torsors, and for each v, a map of Qp-schemes loc : H1(G ,U ) H1(G ,U ) v T n → v n 1 1 1 in terms of which we define Hf (GT ,Un) to be the preimage locp− (Hf ) of 1 2 Hf (Gp,Un) under locp. These fit into commuting squares like so. (Z) / (Z ) X X v

j jv 1 / 1 Hf (GT ,Un) H (Gv,Un) locv The vertical map j is called the global unipotent Kummer map, and its local counterpart jp is called the local unipotent Kummer map. As above, we refer the reader to §2 of Selmer varieties [Kim3] for the details of these constructions.

2.5. Proposition. Let v be a prime = p. Then the subset Im jv of the 1 6 rational points of H (Gv,Un) is ﬁnite. Proof. See Kim-Tamagawa [KT].

2.6. Remark. For v / T a prime of good reduction, we have Im jv = 0; see the proof of Corollary∈ 0.3 in §2 of loc. cit.

2 Technically speaking, while the map locv appearing in the diagram is a morphism of Qp-schemes, the vertical maps j, jv are just maps of sets into the sets of Qp-points of the varieties below. 10 BALAKRISHNAN,DAN-COHEN,KIM,ANDWEWERS

2.7. Definitions. We define the Selmer scheme of to be the (infinite) intersection X n 1 Sel ( ) := loc− Im j X v v v\=p 6 with scheme structure defined by the sum of the corresponding ideals. We also refer to H1(G ,U ) as the local Selmer scheme of near p. As n varies, f p n X these form two towers: . . . .

locp Sel2( ) / H1(G ,U ) X f p 2

locp Sel1( ) / H1(G ,U) X f p compatible with the maps locp as well as j and jp. Thus, if we set 1 n (Z ) := j− loc (Sel ( ) , X p n p p X we obtain a non-increasing sequence of refinements (Z ) (Z ) (Z ) (Z) X p ⊃X p 1 ⊃X p 2 ⊃···⊃X of the set of Zp-points, containing the set of global points. We say that p-adic points which are contained in (Zp)n are cohomologically global of level n, or weakly global of level n. X 2.8. Our first task is to remove the apparent dependence on T . Lemma. Let Γ and U be topological groups with Γ acting continuously on U. Let N Γ be a closed normal subgroup. Then there is an exact sequence of pointed⊂ sets 1 H1(Γ/N, U N ) i H1(Γ,U) r H1(N,U). → −→ −→ Proof. Recall that continous cohomology is defined ([Kim2], section 1) as H1(Γ,U)= U Z1(Γ,U), \ where Z1(Γ,U) consists of the continuous maps c :Γ U such that → c(g1g2)= c(g1)g1c(g2), 1 1 while (uc)(g)= uc(g)g(u− ) for u U and c Z (Γ,U). It is clear that r i sends everything∈ to the base-point.∈ Assume r(c) = 0 for ◦ 1 a continuous cocycle c :Γ U. So there is a u U such that c(n)= un(u− ) for all n N. Define → ∈ ∈ 1 b(g)= u− c(g)g(u), a cocycle in the same U-orbit as c. Then 1 1 1 b(n)= u− c(n)n(u)= u− un(u− )n(u)= e ANON-ABELIANCONJECTURE 11 for all n N. Thus, ∈ b(gn)= b(g)gb(n)= b(g)g(e)= b(g) 1 for all g Γ and n N. Since this also implies b(ng) = b(gg− ng) = b(g), we get ∈ ∈ nb(g)= b(n)nb(g)= b(ng)= b(g) for all g Γ and n N. That is, b factors to a cocycle ∈ ∈ ¯b :Γ/N U N , → which is continuous since Γ/N has the quotient topology.

Proposition. If T ′ and T are two ﬁnite sets of primes that contain all primes of bad reduction and p, the natural restriction maps 1 1 1 (Hf (GT ,Un) ֒ Hf (GT T ′ ,Un) ֓Hf (GT ′ ,Un → ∪ ← induce isomorphisms of Selmer schemes. Proof. We need only consider an enlargement of T to T T . We work with ′ ⊃ points with values in an arbitrary Qp-algebra, which we will omit from the notation. We will provisionally put the sets of primes into the notation, as n n n in Sel ( ). Clearly Sel ( ) ֒ Sel ′ ( ). Recall that T contains already all T X T X → T X primes of bad reduction and p. In particular, the action of Gv for every prime v T T on U is unramiﬁed. Thus, the image of (Z ) in H1(G ,U ) ∈ ′ \ n X v v n is trivial (§2.5). That is, when v T ′ T , for a cohomology class in c 1 ∈ \ ∈ H (GT ′ ,Un), the condition of locally belonging to the image of jv is actually the same as triviality at v. Thus, c goes to zero under any of the restriction maps 1 1 H (G ′ ,U ) H (I ,U ). T n → v n Since the Iv act trivially on Un, this implies that c goes to zero under the restriction map 1 1 H (G ′ ,U ) H (N,U ), T n → n where N GT ′ is the subgroup generated by Iv for v T ′ T . According to ⊂ 1 ∈ \ lemma 2.8, it follows that c comes from H (GT ,Un). By the commutativity of the triangle 1 1 H (G ,U ) / H (G ′ ,U ) T ❖n T n ❖❖❖ ♦♦♦ ❖❖❖ ♦♦♦ ❖❖ ♦♦♦ locv ❖❖' w♦♦♦ locv 1 H (Gv,Un) the local conditions remain the same for both spaces, and hence, n n Sel ( ) Sel ′ ( ). T X ≃ T X Corollary. The subset (Zp)n (Zp) is independent of the choice of the set of primes T . X ⊂X 12 BALAKRISHNAN,DAN-COHEN,KIM,ANDWEWERS

2.9. Now we consider the possibility of a change of base-point from b to c. For this discussion, we will write U(b) and U(c) for the prounipotent p-adic étale fundamental groups with base-points at b and c respectively. Denote by P (b, x) the torsor of prounipotent p-adic étale paths from b to x (P (x) above). Now, given any torsor W for U(b), we get the torsor W c := W P (b, c) = [W P (b, c)]/U(b). ×U(b) × 1 Here the action of u U(b) takes (w, γ) W P (b, c) to (wu, u− γ). This construction defines a∈ map from the groupoid∈ of×U(b) torsors to the groupoid of U(c)-torsors. Lemma. If b and c are both integral, then W W c 7→ maps unramified torsors at v / T to unramified torsors, and crystalline torsors at p to crystalline torsors.∈ Proof. The condition of being unramified at v is given by triviality under the restriction map H1(G ,U) H1(I ,U), v → v while the crystalline condition is given by triviality under the map H1(G ,U) H1(G ,U(B )). p → p cr But since P (b, c) is itself unramified at v / T and crystalline at p, both conditions are preserved by the functor. ∈ That is, we are assured of an isomorphism ( )c : H1(G ,U(b)) H1(G ,U(c)). · f T ≃ f T Meanwhile, since (P (b, x))c = P (c, x), torsors of paths are preserved under the functor. So we conclude Proposition. The functor W W c induces isomorphisms of local and global Selmer schemes commuting7→ with the corresponding localization maps locp and Kummer maps j and jp.

Corollary. The subset (Zp)n (Zp) is independent of the choice of base-point b. X ⊂ X 3. The Conjecture and its context 3.1. We preserve the situation and notation of §1. In particular, denotes a minimal Z-model of a hyperbolic curve over Q as in Segment 2.1.X In his lectures at the IHÉS in February of 2012, M.K. proposed the following. Conjecture. Equality (Z ) = (Z) is obtained for large n. X p n X ANON-ABELIANCONJECTURE 13

3.2. Remark. Although we would expect a suitable generalization of our conjecture to hold over general number ﬁelds, the exact statement is not entirely clear, and we do not go into this issue in this paper. See Dan-Cohen [DC] for the case of the thrice punctured line.

3.3. Recall that j, jp denote the global and local Kummer maps, respectively (2.4). Alongside conjecture 3.1, we consider the following statements. (SGK) Surjectivity of the global Kummer map. The global Kummer map j defines a surjection X(Z) ։ P Seln( ) loc (P ) Im j ∈ X p ∈ p onto the set of torsors which are geometric everywhere locally, for large n. (ILK) Injectivity of the local Kummer map. Suppose x (Z) and y (Z ). If j (x)= j (y) for all n then x = y. ∈ X ∈ X p p p Trivially, we have the implications SGK + ILK Conjecture 3.1 ILK. ⇒ ⇒ 3.4. Relationship to Tate–Shafarevich and Section Conjectures. We now discuss the relationship between 3.3(SGK), finiteness of Sha, and the Grothendieck section conjecture. Let X be a proper hyperbolic curve over Q, let b X(Q), fix an algebraic closure Q¯ of Q, and let GQ denote the Galois ∈ ¯ ˆ ét group of Q/Q. For each x X(Q) we let P (x) denote the π1 (XQ¯ , b)-torsor associated to x. Recall that∈ the Grothendieck section conjecture states that ˆj : x Pˆ(x) 7→ defines a bijection 1 ét X(Q)= H GQ,π1 (XQ¯ , b) . The surjectivity of ˆj bears an obvious relationship to statement 3.3(SGK). ét When we replace π1 (XQ¯ , b) by its prounipotent completion U, the cohomology set becomes a positive dimensional variety, so surjectivity ceases to be plausible; j may nevertheless surject onto those cohomology classes which are everywhere locally geometric. This is motivated in part by the case of elliptic curves and the conjectured finiteness of Ш, through the following basic proposition. 3.4.1. Let E be an elliptic curve over Q. As above, we fix a decomposition group G GQ at every prime v, giving rise to a localization map v ⊂ 1 ét 1 ét loc : H GQ,H (E¯ , Q ) H G ,H (E¯ , Q ) . v 1 Q p → v 1 Q p 14 BALAKRISHNAN,DAN-COHEN,KIM,ANDWEWERS

Qp Qp Let j , jv denote the global and local Qp-linearized Kummer maps, as in the following square.

E(Q) Z Q / E(Q ) Z Q /p ⊗ p p v /p ⊗ p p Qp Qp j jv 1 1 H GQ,H1(EQ¯ , Qp) / H Gv,H1(EQ¯ , Qp) Here the subscript /p denotes p-adic completion. Proposition. Suppose the p-part of Ш(E) is ﬁnite. Then the global (abelian) Qp Qp-linearized Kummer map j deﬁnes a bijection

1 ét Qp E(Q) Z Q = P H GQ,H (E¯ , Q ) loc (P ) Im jv for all primes v ⊗ p p ∈ 1 Q p v ∈ between the vector space of linear combinations of rational points and the classical p-adic Selmer group. Proof. The ﬁniteness of the p-part of Ш implies that the product of the localization maps induces an injection 1 n 1 n . lim H (GQ, E)[p ] Q ֒ lim H (G , E)[p ] Q (*) ⊗ p → v ⊗ p ←− Yv ←− Recall that there’s a Galois-equivariant isomorphism Q lim E[pn](Q)= Hét(E , Q ). p ⊗ 1 Q p We consider the inverse system←− of short exact sequences pn 0 / E[pn] / E / E / 0 O O p

0 / E[pn+1] / E / E / 0. pn+1

Taking Q-valued points followed by invariants by GQ, we obtain an inverse system of short exact sequences

n 1 n 1 n 0 / E(Q)/p / H GQ, E[p ](Q) / H GQ, E(Q) [p ] / 0 O O O

n+1 1 n+1 1 n+1 0 / E(Q)/p / H GQ, E[p ](Q) / H GQ, E(Q) [p ] / 0. Taking inverse limits and tensoring with Qp, we obtain the top row in the following diagram:

Qp j 1 ét 1 n 0 / E(Q) Q / H GQ,H (E , Q ) / lim H GQ, E(Q) [p ] Q / 0 /p ⊗ p 1 Q p ⊗ p ←− / / 1 ét / 1 n / 0 E(Qv)/p Qp H GQv ,H1 (EQ , Qp) lim H GQv , E(Qv) [p ] Qp 0; ⊗ Qp v ⊗ jv ←− ANON-ABELIANCONJECTURE 15

repeating the procedure with Qv in place of Q gives us the rest of the diagram. Varying the place v and using the injectivity (*), we obtain an exact sequence like so 1 ét Q 1 ét H GQv ,H1 (EQ, p) 0 E(Q) Qp H GQ,H (E , Qp) . → /p ⊗ → 1 Q → E(Q ) Q Yv v /p ⊗ p By the Mordell-Weil theorem, we have E(Q) Q = E(Q) Q , /p ⊗ p ⊗ p so the proposition follows. 1 1 Corollary. We have E(Q) Qp = Sel (E). In particular, our Sel (E) is equal to the classical p-adic⊗ Selmer group. Proof. For v = p, the geometricity condition 6 loc (P ) Im j v ∈ v is actually equivalent to the (a priori weaker) condition

Qp loc (P ) Im jv v ∈ since E(Q ) Q = 0. v /p ⊗ p Qp On the other hand at v = p the condition loc (P ) Im jp is equivalent to p ∈ locp(P ) being crystalline according to Example 3.11 of Bloch-Kato [BK]. 4. The unipotent Albanese map and local height on elliptic curves 4.1. Setup and statement. 4.1.1. As explained in the introduction, our goal here is to investigate a relation between local heights and Albanese maps, with a view towards ap- plying it to the computation of some simple Selmer schemes. The relation over a finite extension of Qp was noticed earlier following the paper [BKK] by its authors and Amnon Besser [BB]. The main purpose here will be to work out a precise relation over Q for l = p, when the curve has bad reduction. l 6 4.1.2. Fix an odd prime p, let Fv be an unramified finite extension of Ql for l = p, and let (Ev, O) be an elliptic curve over Fv written in Weierstrass minimal6 form 2 2 3 2 2 3 Z0Z2 + a1Z0Z1Z2 + a3Z0 Z2 = Z1 + a2Z0Z1 + a4Z0 Z1 + a6Z0 . Let X = E O with equation v \ { } 2 3 2 y + a1xy + a3y = x + a2x + a4x + a6. We let b be the tangent vector to E at O dual to the invariant differential form dx/(2y + a1x + a3), 16 BALAKRISHNAN,DAN-COHEN,KIM,ANDWEWERS which we will use as the main base-point for fundamental groups. Let z = ( x/y), which is a b-compatible uniformizing element at O in that (d/dz) O = b−. (We refer to Silverman [Sil3], chapter 4, for this and other assertions about| the coordinates on the Weierstrass minimal model.) 4.1.3. Let log denote the p-adic logarithm normalized so that log(p) = 0. The p-adic logarithm ab log χ : Gv Qp → 1 of the p-adic cyclotomic character may be regarded as an element of H (Gv, Qp). Recall that the cup product defines a Qp-valued pairing H1(G , Q ) H1(G , Q (1)) H2(G , Q (1)) ∼= Q . v p × v p → v p −→ p Let rec denote the reciprocity map of abelian class field theory ab F ∗ G v → v and recall that l denotes the residue characteristic of Fv. Then for a Fv∗ we have the formula ∈ log χ k(a) = log(χ(rec (a))) = log lv(a) = log a . ∪ v − | |v 4.1.4. Proposition. We have 0 1 2 H (Gv,U1)= H (Gv,U1)= H (Gv,U1) = 0. Proof. We start with H0: by the weight–monodromy theorem, proved for abelian varieties by Grothendieck in [SGA, Exposé IX], the inertia fixed part of E[pn] has Frobeinus weight 2, so in particular has no Frobenius-fixed part, whence the vanishing. The− vanishing of H2 then follows by local Tate duality [NSW], since Vp(Ev) is self-dual. Since the v-adic absolute value of pn is 1, it follows from [NSW, Theorem 7.3.1] that the Euler characteristic is zero; combined with the vanishing of H0 and H2, this implies the vanishing of H1.

4.1.5. Recall that the Gv-equivariant extension 1 Q (1) U U 1 → p → 2 → 1 → gives rise to an exact sequence of pointed sets H0(U ) H1(Q (1)) α H1(U ) H1(U ). 1 → p −→ 2 → 1 The vanishing of the extreme terms implies that α is bijective, so that we can choose a cocycle representing jv(x) which takes values in Qp(1). Thus, we get a function φ : X(F ) Q v v → p via the formula φ (a) = log χ j (a). v ∪ v ANON-ABELIANCONJECTURE 17

4.1.6. We deﬁne a p-adic local Néron function to be a function λ : E (F ) Q v v → p which satisﬁes the following properties with respect to the coordinates x, y.

(i) λ is continuous on Ev(Fv) O and bounded on the complement of any v-adic neighborhood of\ {O. } (ii) The limit 1 lim λ(a) log x(a) v a 0 − 2 | | exists. → (iii) For all a E (F ) with [2]a = 0, ∈ v v 6 λ([2]a) = 4λ(a) log (2y + a x + a )(a) . − | 1 3 |v Theorem. The function φv is a p-adic local Néron function. The proof of theorem 4.1.6 appears in segment 4.4 below.

4.2. Construction of π[2]-tower.

4.2.1. We write here π[2] for π(p)(X,¯ b)/[π(p)(X,¯ b), [π(p)(X,¯ b),π(p)(X,¯ b)]], the quotient of the pro-p fundamental group of X¯ = X F¯v by the third level of its lower central series. We will need to consider differen⊗ t base-points w below, in which case we denote the group by π[2](w). Similarly, the pushout to π[2](w) of the homotopy class of maps from w to y will be denoted by π[2](w,y): (p) ¯ π[2](w,y) := π1 (X; w,y) (p) ¯ π[2](w). ×π1 (X,w) Note that when b is replaced by λb for λ Fv, then the compatible uniformizer is changed to z/λ. ∈ We note that π[2] fits into an exact sequence 0 Z π T E 0 → → [2] → p → where Z Z (1) ≃ p is generated by [e, f] for any lift e, f of a basis for TpE. As in Lemma 1.1 of [Kim4], this exact sequence{ has} a Galois-equivariant splitting which extends also to a splitting of the sequence 1 Q (1) U V E 0. → p → 2 → p → Thus, as in [Kim4, p. 730], we will write a cocycle c : G U v → 2 as c = c2c1, 18 BALAKRISHNAN,DAN-COHEN,KIM,ANDWEWERS where c2 takes values in Qp(1), c1 is a cocycle with values in VpE, and dc = (1/2)c c . 2 − 1 ∪ 1 We wish to compute the group π[2] using theta groups. The result is stated in proposition 4.2.10 below. n n 4.2.2. Let D0 := [p ]∗[O], the sum of all points of Ev[p ]. We write for linear equivalence of divisors. We claim that ∼ D p2n[O]. 0 ∼ To see this we base change to an algebraically closed field, write p2n 1 − D0 = [zj], Xj=0 and remember the isomorphism of group schemes E ∼ Pic0 E v −→ v z [z] [O], 7→ − from which D p2n[O]= [z ] p2n[O] 0 − j − Xj = [z ] [O] j − Xj z [O] ∼ j − h Xj i = [O] [O] − = 0. Let := (pn[O]). Then we have Hn O n p pn pn[O] = p2n[O] (D ), Hn ≃ O O ≃ O 0 via an isomorphism well-defined up to a constant. 4.2.3. In general, an isomorphism (A) (B) O ≃ O must be defined by a rational function f such that (f)= B A, which takes a section s (A) and multiplies it by f. We will denote this− isomorphism also by f: ∈ O f (A) (B). O ≃ O When a tangential base-point w at O has been chosen we normalize all such isomorphisms as follows. Choose a local coordinate t at O so that (d/dt) = w. Then we normalize f so that |O ord (f) t− O f(O) = 1. ANON-ABELIANCONJECTURE 19

When this normalization has been fixed, we will refer to the function or the isomorphism as based at w. This way, when f g (A) (B), (B) (C) O ≃ O O ≃ O and h (A) (C), O ≃ O are all based at w, then we can be sure that h = gf. We will be able to deduce the commutativity of various diagrams using this fact. The based function giving the isomorphism n p (D ) Hn ≃ O 0 given a base-point w will be denoted by fw. More generally, given any function g such that (g)= A B − we will write gw for the constant multiple of g that is based at the tangent vector w. 4.2.4. For our choice of tangent vector b, an elementary computation shows that the function y is based, that is, y z3. In the case of the function ∼ Z fb Q[x,y], clearly, there is a constant multiple f Z[x,y] with the property∈ that (f Z) = (p2n 1)[O] on the Weierstrass∈ minimal model. But then, since (z)∞ = ( x/y)− = [O]+ D on the minimal model with D − 1 p2n Z disjoint from [O], we see by comparing divisors that z − f is a unit h in a neighborhood of the section O. Thus, its value on O is a unit u v∗. Hence we see that ∈ O 1 Z 1 p2n 1 fb = u− f = u− hz − , with the second equality holding in a neighborhood of O. In particular, the formal power series expansion of f in the parameter z has coefficients in . b Ov 4.2.5. Define the subscheme

X′ n ⊂Hn as the inverse image of the section 1 Γ( (D0)) under the map n ∈ O ( )p n · p (D ), Hn −−−→Hn ≃ O 0 where the second isomorphism is given by the function fb/pn . Standard Kummer theory implies that

X′ E n → v is a finite µpn cover (totally) ramified only over D0. In particular, pn r : X′ E E , n n → v −→ v is a finite cover. 20 BALAKRISHNAN,DAN-COHEN,KIM,ANDWEWERS

1/pn 4.2.6. Since the cover X is constructed locally as [(f n ) ] and n′ OE b/p p2n 2 3 (u f n )= u + c u + c u + b/p 2 3 · · · formally with respect to the uniformizer u = pnz, we see that the tangent n vector b/p = (d/du) O lifts to a tangent vector b′ to Xn′ at the unique point above O. That is, r| : X X is equipped with an F -rational lift of n n → v the tangential base-point b. Therefore, there is a Gv-equivariant surjective homomorphism π (X ) [2] → n b that sends the identity to the base-point lift b′. Here the subscript ( )b refers to the tangential ﬁber functor of Deligne [Del, §15]. We will use theta· groups to show that this map induces a bijection π ∼ lim(X ) . [2] −→ n b ←− 4.2.7. For each x E , we let τ : E E be the translation operator ∈ v x v → v τx(y) = y + x. Recall from section 23 of Mumford [Mum] that the theta group ( ) G Hn associated to is the group scheme over F whose R-points, for R an Hn v Fv-algebra, are commuting squares g n,R / n,R H ∼= H

E ∼= / E v,R τx v,R for x Ev(R). Since x is determined by g, we denote such a square simply by g.∈ If x E (R) then the ideal deﬁning the associated closed subscheme ∈ v of Ev,R is locally principal; we denote the associated Cartier divisor by [x]. In this notation, we have n τ ∗ x n = (p [x]) − H ∼ O n n isomorphic to (p [O]) if and only if x Ev[p ](R). The theta group therefore ﬁts intoO a short exact sequence ∈ ρ 0 G ( ) E [pn] 0. → m → G Hn −→ v → Moreover, after forgetting the group structures, the projection ρ admits a section (see segment 4.3.1 below). We will also consider a point g ( ) as an isomorphism ∈ G Hn g : ∼ τ ∗ , Hn −→ ρ(g)Hn in which case composition in ( ) is given by the formula G Hn (*) g h = τ ∗ (g) h. · ρ(h) ◦ ANON-ABELIANCONJECTURE 21

4.2.8. Taking tensor powers deﬁnes a map of exact sequences (solid arrow diagram below)

pn 2n 0 / G / ( n ) / E [p ] / 0 Om G HO h v n pn ( )⊗p S · β 0 / G / ( ) / E [pn] / 0. m G Hn ρ v We now construct a diagonal homomorphism β, as shown, which will make n the triangle to its upper right commute. Given x Ev[p ] we let ψx denote the canonical isomorphism ∈

(D ) ∼ τ ∗ (D ). O 0 −→ x O 0 (which is not necessarily compatible with the base point). We note that

(*) ψO = Id (D0), O and that if y E [pn] is a second point, then the square ∈ v ∗ ψ τ (ψx) y / y / (**) (D0) ❱ τy∗ (D0) τy∗τx∗ (D0) O ❱❱❱❱ O O ❱❱❱❱ ❱❱❱❱ ψ ❱❱❱❱ x+y ❱❱❱❱* τ (D ) x∗+yO o commutes. We deﬁne β by 1 β(x) := τ ∗f ψ f − . x ◦ x ◦ Properties 4.2.7(*), 4.2.8(*), and 4.2.8(**) combine to show that β is a homomorphism: 1 β(O)= f Id f − = Id, ◦ ◦ and 1 1 β(x) β(y)= τ ∗(τ ∗f ψ f − ) τ ∗f ψ f − · y x ◦ x ◦ ◦ y ◦ y ◦ 1 1 = τ ∗ f τ ∗(ψ ) τ ∗f − τ ∗f ψ f − x+y ◦ y x ◦ y ◦ y ◦ y ◦ 1 = τ ∗ f τ ∗(ψ ) ψ f − x+y ◦ y x ◦ y ◦ 1 = τ ∗ f ψ f − x+y ◦ x+y ◦ = β(x + y).

4.2.9. We deﬁne pn 1 := ( )⊗ − Im β . Gn · This subgroup of the theta group ﬁts into an exact sequence ρ n ( ) 0 µ n E [p ] 0. ∗ → p → Gn −→ v → As is customary when doing Galois-theoretic computations, we will often identify with (F¯ ). Gn Gn v 22 BALAKRISHNAN,DAN-COHEN,KIM,ANDWEWERS

Lemma. The ﬁnite étale covering r : X X n n → is Galois with Galois group . Gn Proof. We claim that the square pn ( n ) G HO d■ ■■ β n ■■ p ■■ ⊗ ■■■ ( ) E [pn] n 9 v G H tt tt S tt tt ρ tt Gn commutes. This is an elementary computation which we carry out anyway. We put ourselves in the general setting of a diagram ρ / GO a❇ HO ❇❇ ❇❇ t ❇❇ u β ❇ ? G′ / H′ ρ′ in which the outer square and the upper right triangle commute, and u is injective as shown. If g t 1(Im β) then there’s a g G such that ∈ − ′ ∈ ′ t(g)= β(ρ′(g′)). We then have uρ′g′ = ρβρ′g′ = ρtg = uρ′g, from which ρ′g′ = ρ′g, so that t(g)= βρ′g′ = βρ′g, as hoped. It follows that the diagram

pn Hn H

E [pn] Gn v commutes, in the sense that the map of G-sets is linear over the map of groups. n Denote by Yn the image of Ev Ev[p ] under the section 1 viewed as a map of schemes \ E (D ). v → O 0 ANON-ABELIANCONJECTURE 23

n Since the action of Ev[p ] on (D0) given by the liftings ψx maps a function h (viewed as a section) to h Oτ , Y is stable under the action of E [pn] via ◦ x n v β. Hence, n acts on Xn. Further,G pn n n ∼= (D0) H →H O pn is a µpn -torsor away from the zero section, where this µ is exactly the n subgroup of mapping to 1 under the map ( ) p . Therefore, X is a Gn · ⊗ n µpn -torsor over Yn and Xn/µpn ∼= Yn. n However, Yn is isomorphic to Ev Ev[p ] equivariantly with respect to the n \ action of Ev[p ]. So X / = Y /E [pn] = E O = X, n Gn ∼ n v ∼ v \ which completes the proof of the lemma.

¯˜ ¯ 4.2.10. If we denote by X[2] X the quotient of the universal covering ¯ → space of X corresponding to π[2], there is a surjective homomorphism (*) Aut (X¯˜ /X¯ ) ։ , [2] Gn n simply because n is a central extension of Ev[p ]. The signiﬁcance of the theta-group for usG is that the commutator map

[ , ] : µ n · · Gn × Gn → p factors to the Weil pairing n n , : E [p ] E [p ] µ n , h· ·i v × v → p so is in particular surjective (this is a well known fact, explained for instance in [MvdGE, Chapter XI, Proposition 11.20]). Hence, we also have a surjection ¯˜ ¯ ¯˜ ¯ ։ [Aut (X[2]/X), Aut (X[2]/X)] µpn . Proposition. The surjections 4.2.10(*) induce an isomorphism of profinite groups Aut (X¯˜ /X¯) lim , [2] ≃ Gn and hence, a bijection of profinite sets ←− π lim(X ) . [2] ≃ n b ←− Proof. It suffices to show injectivity. So let g Aut (X¯˜ /X¯) be non-trivial. ∈ [2] ¯˜ ¯ ab If g has non-trivial image in Aut (X[2]/X) TpE, then clearly there is a map ≃ Aut (X¯˜ /X¯) [2] → Gn which does not send it to zero. So assume g [Aut (X¯˜ /X¯ ), Aut (X¯˜ /X¯ )]. ∈ [2] [2] 24 BALAKRISHNAN,DAN-COHEN,KIM,ANDWEWERS

But then, since [Aut (X¯˜ /X¯ ), Aut (X¯˜ /X¯ )] Z (1) [2] [2] ≃ p as a topological group, the family of surjections ¯˜ ¯ ¯˜ ¯ ։ [Aut (X[2]/X), Aut (X[2]/X)] µpn must be separating. For the statement about π[2], recall that the formula

b′l = φ(b′) for l π and φ Aut (X¯˜ /X¯) deﬁnes an anti-isomorphism from π to ∈ [2] ∈ [2] [2] ¯˜ ¯ Aut (X[2]/X).

4.3. Interaction of local Kummer map with multiplication by 2. Our goal here is to derive an explicit formula for the change in jv(x) when we multiply x by 2. The result is stated in corollary 4.3.8. 4.3.1. We can construct a canonical section of the surjection ρ (4.2.9( )) as follows. Notice that the automorphism ∗ [ 1] : E E − v ≃ v lifts to an automorphism [ 1] : − Hn ≃Hn that sends a section φ(y) to φ( y). This induces an involution − i : Gn → Gn that sends g to [ 1] g [ 1]. − ◦ ◦ − n Lemma. If R is an Fv-algebra, then any R-valued point of Ev[p ] has an R-valued half. Proof. We recall the proof of this well-known fact. There’s a short exact sequence of ﬁnite étale group schemes 1 n n 0 E [2] [2]− E [p ] E [p ] 0, → v → v → v → hence an exact sequence of étale cohomologies 1 n n δ 1 [2]− E [p ](R) E [p ](R) H (Spec R, E [2]). v → v −→ v The boundary map δ is a map from a Z/pn-module to a Z/2-module, hence, under our assumption that p is odd, necessarily zero. Given any element x E [pn] and a lift g of x/2, the element ∈ v ∈ Gn − 1 x′ := i(g)g− is independent of the lift g, and can be characterized as the unique element of 1 n lying over τx that is anti-symmetric with respect to i, in that i(g) = (g)− . G Thus, we can write an element g uniquely in the form ∈ Gn g = g2g1 ANON-ABELIANCONJECTURE 25

1 where g1 is the unique lift of ρ(g) satisfying i(g1) = g1− . Sometimes we abuse notation and write g1 both for this lift and for ρ(g). 1 4.3.2. Recall that the nonabelian cohomology set H (Gv,π[2]) can be constructed as a set of equivalence classes of continuous cocycles G π ; v → [2] it also parametrizes the set of isomorphism classes of Gv-equivariant π[2]- torsors. Given a point x X(F ) we write ˆj(x) for the associated class ∈ v ˆ (p) ¯ 1 j(x) := [π1 (X; b, x) (p) ¯ π[2]] H (Gv,π[2]). ×π1 (X,b) ∈ If cx is an associated cocycle, then composing with the anti-homomorphism π ։ , [2] Gn we obtain an anti-cocycle G . v → Gn which we continue to denote by cx. Explicitly, cx is constructed as follows. n We choose a point y X(F¯v) such that p y = x, and a point z Xn(F¯v) lying above y. Then for∈ γ G , cx(γ) is determined by the formula∈ ∈ v ∈ Gn γ(z)= cx(γ)(z). We remark that the anti-cocycle condition is given by x x x c (γ1γ2) = (γ1c (γ2))c (γ1). Using the section of ρ constructed in segment 4.3.1, we can canonically de- compose cx as x x x c = c2 c1 x x x with c2 taking values in µn and c1 the anti-symmetric element lifting ρ(c ). 4.3.3. We now ﬁx several isomorphisms of line bundles relating to multiplication by 2 and by pn. There are isomorphisms

[2]∗( ([O])) ([O] + [x ] + [x ] + [x ]) (4[O]); O ≃ O 1 2 3 ≃ O the ﬁrst isomorphism is canonical, since

[2]∗(O) = [O] + [x1] + [x2] + [x3] while for the second isomorphism we may take the one induced by the function 2y + a x + a h = 1 3 . b 2 That is, this function has divisor 3[O] ([x1] + [x2] + [x3]) and is compatible with the tangent vector b. − There is an isomorphism n n [2]∗( ) = [2]∗ (p [O]) (p ([O] + [x ] + [x ] + [x ])) Hn O ≃ O 1 2 3 (4pn[O]), ≃ O with the ﬁrst two isomorphisms being canonical while the third we take to be given by pn (hb/pn ) . 26 BALAKRISHNAN,DAN-COHEN,KIM,ANDWEWERS

There is an isomorphism

[2]∗ (D ) (D + D + D + D ) (4D ) O 0 ≃ O 0 1 2 3 ≃ O 0 n with the last isomorphism being induced by the function hb [p ]. Finally, an isomorphism ◦ n (4pn[O])p (4D ) O ≃ O 0 is induced by f 4 . Given x E E [2], choose y such that pny = x. b/pn ∈ v \ v Taking ﬁbers above the points y and 2y, we obtain a commutative diagram

pn 4 ( ) n fb/pn 4 · 4p ( n)y / ( n)y / (4D0)y H H ∼= O

n 2n −p −p n −1 = h = h n = (hb p ) ∼ b/pn ∼ b/p ∼ ◦

pn ( ) n f2b/pn [2] p ◦ ([2]∗ n)y · / ([2]∗ n )y / ([2]∗ (D0))y H H ∼= O

= ∼= ∼ ∼=

pn ( ) n f2b/pn · / p / ( n)2y ( n)2y (D0)2y H H ∼= O where the lower vertical arrows are all natural base-change maps. The maps induced by functions have all been based so as to make all diagrams commutative. The maps are also clearly compatible with the action of the Galois group Gv.

4.3.4. We consider now the relation between the action of g n and the composition of the leftmost vertical isomorphisms in the diagram,∈ G which we will denote by B(y) : ( )4 ( ) . Hn y ≃ Hn 2y In the following, we will give the argument pointwise over E, even though the underlying discussion is about the corresponding scheme isomorphism 4 B : ( ) [2]∗( ). Hn ≃ Hn Lemma. There is a commutative diagram

g4 4 1 / 4 ( n)y ( n)ρ(g)(y) H ∼= H B(y) ∼= ∼= B(ρ(g))(y) 2g1 ( n)2y / ( n)(2ρ(g))(2y) H ∼= H where we denote by 2g1 the anti-symmetric lift of the element 2ρ(g). ANON-ABELIANCONJECTURE 27

Proof. We consider the isomorphism 4 1 B(ρ(g)(y)) g⊗ B(y)− : ( ) ( ) ◦ 1 ◦ Hn 2y ≃ Hn (2ρ(g))(2y) lifting the action of 2ρ(g). We need only check that 4 1 4 1 1 i(B(ρ(g)(y)) g⊗ B(y)− )= B( y) [(g )⊗ ]− B( ρ(g)(y))− . ◦ 1 ◦ − ◦ 1 − For this, we embed the previous diagram into the bigger diagram

[ 1] g4 [ 1] 4 − / 4 1 / 4 − / 4 ( n) y ( n)y ( n)ρ(g)(y) ( n) ρ(g)(y) H − ∼= H ∼= H ∼= H − = B( y) = B(y) = B(ρ(g)(y)) = B( ρ(g)(y)) ∼ − ∼ ∼ ∼ − [ 1] 2g [ 1] − 1 − ( n) 2y / ( 2)2y / ( n)(2ρ(g))(2y) / ( n) (2ρ(g))(2y). H − ∼= H ∼= H ∼= H − The two squares on the left and right are clearly commutative. But 4 1 [ 1] B(ρ(g)(y)) g⊗ B(y)− [ 1] − ◦ ◦ 1 ◦ ◦ − 4 1 = B( ρ(g)(y)) [ 1] g1⊗ [ 1] B( y)− − ◦ − ◦ 1 ◦4 − ◦ 1 − = B( ρ(g)(y))(g1− )⊗ B( y)− − 4 1 − 1 = B( ρ(g)(y))(g⊗ )− B( y)− − 1 − at every y as desired.

n 4.3.5. Choose y as above so that p y = x and let v Xn lie above y. Then for γ G , we have ∈ ∈ v x x x γ(v)= c (γ)v = c (γ)c (γ)v ( ) x . 2 1 ∈ Hn ρ(c (γ))y Hence, 4 4 x 4 x 4 γ(v⊗ ) = (γ(v))⊗ = (c2 (γ)) (c1 (γ)v)⊗ . (Recall from segment 4.2.8 that tensor powers restrict to ordinary powers in µpn .) Hence, 4 4 x 4 x 4 γ(B(y)(v⊗ )) = B(γ(y))((γv)⊗ )= B(γ(y))((c2 (γ)) (c1 (γ)v)⊗ ) which by Lemma 4.3.4 x 4 x 4 = (c2 (γ)) (2c1 (γ))(B(y)(v⊗ )). 28 BALAKRISHNAN,DAN-COHEN,KIM,ANDWEWERS

4.3.6. We use the diagram of segment 4.3.3 to ﬁnd that the map (D ) H2y → O 0 2y 4 sends B(y)(v⊗ ) to n 1 1 (hb(p y))− (1 (D0))(2y) = (hb(x))− (1 (D0))(2y). O O

Therefore, an element in the inverse image (Xn)2y of (1 (D0))(2y) is O p−n 4 (hb(x)) B(y)(v⊗ ). n Therefore, if we let k( ) n denote the mod p abelian Kummer map · p 1 n F ∗ H G , Z/p (1) v → v given in terms of the choice of a (pn)th root by γ ( p√n a) k(a) n (γ) := , p p√n a then the Galois action on this element is given by the cocycle x 4 x k(hb(x))pn (c2 ) (2c1 ). This must be the same as the action via c2x, by the compatibility of the big diagram with the action of Gv. As we take the limit over n, we get the equality 2x x 4 x c = k(hb(x))(c2 ) (2c1 ) at the level of p-adic cocycles. 4.3.7. One last modification is that this calculation has produced the class [π (2b, 2x)] H1(G ,π (2b)), [2] ∈ v [2] 1 which we need to shift back to H (Gv,π[2]) to get the class ˆj(2x). For this, we need to compose with the class of π[2](b, 2b). We claim that this π[2](b)-torsor corresponds to the cohomology class k(2), where k denotes the profinite abelian Kummer map 1 F ∗ H (G , Z (1)). v → v p To see this, let T X := T E 0 O O v \ { } denote the punctured tangent space at the origin. There’s an Fv-rational isomorphism of vector groups A1 ∼ T E −→ O v sending 1 b, hence an isomorphism of schemes 7→ G ∼ T X m −→ O which sends 1 to b and 2 to 2b. The theory of tangential fiber functors gives rise to an associated morphism of fundamental groupoids. In particular, there’s a map Z (1) = π(p)(G , 1) π(p)(X, b) ։ π (b), p 1 m → 1 [2] ANON-ABELIANCONJECTURE 29 and the induced map H1(G , Z (1)) H1(G ,π (b)) v p → v [2] (p) sends the torsor π1 (1, 2) to π[2](b, 2b). A straightforward calculation, carried out in §14 of Deligne [Del], shows that the former is represented by the Kummer cocycle k(2) as claimed. Therefore, Theorem. Let x X(F ), let cx be an associated anticocycle ∈ v G lim v → Gn as in segment 4.3.2, let ←− x x x c = c2 c1 x x x denote the decomposition of c with c2 taking values in Zp(1) and c1 anti- symmetric (same segment), let hb denote the meromorphic function 2y + a x + a h = 1 3 b 2 of segment 4.3.3, and let k denote the Kummer map. Then x 4 x k(2hb(x))(c2 ) (2c1 ) is an anti-cocycle associated to the point 2x. 4.3.8. We can now push out through the homomorphism π U . [2] → 2 Corollary. Let 1 jv : v(Fv) H (Gv,U2) X → x x be the unipotent Albanese map of level 2 at v. If jv(x) = [c2 c1 ], then x 4 x jv(2x)= k(2hb(x))(c2 ) (2c1 ). 4.4. Proof of theorem 4.1.6.

4.4.1. Lemma. Suppose a X(Fv) reduces to O mod mv = (πv) (the maximal ideal of ). Then∈ there exists an a E (F ) such that OFv ′ ∈ v v n a = p a′.

Proof. Let D(O) denote the residue disk of O inside Ev(Fv). Referring to Silverman [Sil3], Proposition 2.2 of Chapter VII, combined with Example 3.1.3 and Proposition 3.2 of Chapter IV, together provide a bijection D(O) ∼ m , −→ v plus a decreasing filtration of D(O) by subgroups Di(O) compatible with the filtration on mv by powers, such that for each i, the induced map (*) Di(O)/Di+1(O) ∼ mi /mi+1 −→ v v is an isomorphism of groups. Moreover, D(O) is separated and complete with respect to the filtration by the subgroups Di(O). 30 BALAKRISHNAN,DAN-COHEN,KIM,ANDWEWERS

Since F contains Q , l = p, the group m is p-divisible. We may use the v l 6 v group isomorphisms (*) to construct a Cauchy sequence ai in D(O) with pna a mod Di(O). Its limit a is a pnth root of a as hoped.{ } i ≡ ′

4.4.2. Lemma. Suppose a X(Fv) reduces to O mod mv = (πv). Then the anti-cocycle ∈ ca : G (F ) v → Gn v associated to a takes values in µpn . n Proof. According to lemma 4.4.1, a possesses an Fv-rational p th root n a′ Y = E E[p ]. ∈ n \ Let a be an F -point of X lying above a . For γ G , the element ′′ v n ′ ∈ v g := ca(γ) (F ) ∈ Gn v was deﬁned in segment 4.3.2 by

g(a′′)= γ(a′′). Recall from segment 4.2.9 that we have a commutative diagram like so: s Xn Yn

ρ n 0 µ n E [p ] 0. p Gn v Since γ acts trivially on a′, we have

s(a′′)= s(ga′′), and because of the commutativity of the diagram, the latter equals

ρ(g)(s(a′′)). n Since the action of E[p ] on Yn is free, it follows that ρ(g) = 0, hence that g µ n . ∈ p 4.4.3. Near O, in the coordinate z, we have 1 p2n 2 p2n fb = z − + z − g(z) with g(z) [[z]]. Also, ∈ Ov n 2 z(a′) = (1/p )z(a)+ z(a) h(z(a)), for a power series h [[z]]. Therefore, ∈ Ov n 1 p2n n 2 p2n n fb/pn = (p z) − + (p z) − g1(p z) with g (t) [[t]] and 1 ∈ Ov 1 p2n 2 p2n fb/pn (a′)= z(a) − + z(a) − H(z(a)), where H(t) [[t]]. Hence, ∈ Ov 1 p2n fb/pn (a′)= z(a) − u ANON-ABELIANCONJECTURE 31

a for a unit u 1 mod m and (since c takes values in µ n , and since ≡ v p 0 1 k( ) n : H (G ) H (µ n ) · p m → p is a homomorphism to a Z/pn-module) a c = k(fb/pn (a′))pn = k(z(a))pn . 1 Taking the limit over n, we see that in H (Gv, Zp(1)), the class jv(a) is identiﬁed with the Kummer class of z(a). Hence, for a reducing to O mod mv, we have φ (a) = log χ(rec (z(a))) = log(lv(z(a)))= log z(a) = (1/2) log x(a) . v v − | |v | | By this formula, the function φv is bounded on the complement D(O) U \ of any open set U containing O inside the residue disk about O. On the other hand, by Kim-Tamagawa [KT, Corollary 0.2], φ (E D(O)) = φ (X( )) v v \ v Ov is ﬁnite. Thus, φv is bounded on the complement in Ev of any v-adic neighborhood of O. 4.4.4. Finally, by corollary 4.3.8, we have φ (2a) = (log χ) j (2a) v ∪ v = (log χ) k(2h (a))(ca)4 ∪ b 2 = (log χ) k(2h (a)) + 4(log χ) ca ∪ b ∪ 2 = 4φ (a) log 2h (a) v − | b | = 4φ (a) log (2y + a x + a )(a) . v − | 1 3 |v This completes the proof of theorem 4.1.6. 4.5. The range of a p-adic local Néron function.

4.5.1. We temporarily relax our assumption that Fv is unramified over Ql, and let e denote the ramification degree. We normalize our absolute value 1 v by l = l− . When taking p-adic logarithms of absolute values, we may artificially| · | | | define n log(ln/e)= log l. e We also write v = log (a valuation with values in the totally ordered log l − | · | subgroup Z e of Qp), and we write e ord = v. log l 32 BALAKRISHNAN,DAN-COHEN,KIM,ANDWEWERS

Proposition. Suppose the function λ : E (F ) O Q v v \ { }→ p is a p-adic local Néron function in the sense of segment 4.1.6. (a) If a E (F ) reduces to a nonsingular point, then ∈ v v 1 λ(a) = max 0, v(x(a)) . { −2 } (b) Assume E has multiplicative reduction and suppose a E (F ) re- v ∈ v v duces to a singular point. Let N = ord ∆(Ev). Let Ev,0(Fv) denote the group of points reducing to nonsingular points. We choose representatives 0,...,N 1 for Z/N. Then there is a unique isomorphism { − } ( ) n : E (F )/E (F )= Z/N ∗ v v v,0 v such that n(a)(N n(a)) ( ) λ(a)= − log ∆ . ∗∗ 2N 2 | | 4.5.2. For the proof of proposition 4.5.1 we follow the treatment in chapter VI of Silverman [Sil2]. In order to accord with the normalization used there, we set 1 λ′ = λ + v(∆). 12 Then λ′ satisﬁes (i), (ii), and (iii)’ For all a E (F ) with [2]a = 0, ∈ v v 6 1 λ′([2]a) = 4λ′(a)+ v((2y + a x + a )(a)) v(∆). 1 3 − 4 The proof of Theorem 4.1 of loc. cit. applies with the real logarithm replaced by the p-adic logarithm to show that

1 1 1 λ′(a)= max v(x(a)− ), 0 + v(∆), 2 { } 12 which establishes (a). Our proof of (b) is similar; we nevertheless take the time to fill in some details in segments 4.5.3–4.5.4 below. 4.5.3. The proof of Theorem 1.1 of loc. cit. applies with the real logarithm replaced by the p-adic logarithm to show that properties (i)–(iii)’ uniquely determine λ′. Let Lv be a finite extension of Fv of ramification degree e′ over Fv, suppose (b) has been established over Lv, and let λ′ be a function E (F ) O Q v v \ { }→ p which satisfies properties (i)–(iii)’. Then the formulas given in parts (a) and (b) give us a function

λ′ : E (L ) O Q L v v \ { }→ p ANON-ABELIANCONJECTURE 33 which, by uniqueness, extends λ′. Our preferred generator a0 of Ev(Lv)/Ev,0(Lv) gives us a preferred generator e′a0 of Ev(Fv)/Ev,0(Fv). We have

N = NL/e′ and for a E (F ) we set ∈ v v n(a)= nL(a)/e′. Then nL(NL nL) n(N n) −2 log ∆L = −2 log ∆ 2NL | | 2N | | which establishes 4.5.1( ) over F . The uniqueness of 4.5.1( ) follows as ∗∗ v ∗ in Lemma 5.1 of Silverman [Sil1]. So after possibly replacing Fv by a ﬁnite extension, we may assume Ev has split multiplicative reduction.

4.5.4. It follows that Ev is isomorphic to a Tate curve Eq for some q Fv∗ with q < 1 and v(∆) = v(q). Let ψ denote the induced map ∈ | | ։ ψ : Fv∗ Ev(Fv). By Chapter V §4 of Silverman [Sil2], Z = ψ : F ∗/q ∼ E (F ) v −→ v v restricts to = ∗ ∼ E (F ). Ov −→ 0,v v So the isomorphism Ev(Fv)/E0(Fv) ∼= Z/N is realized as F v∗ ord Z/N. qZ −−→ Ov∗ Thus, if a = ψ(u) with 0 < v(u) < v(q), we have n(N n) 1 v(u)2 − log ∆ = v(u) . 2N 2 | | 2 v(q) − So 4.5.1( ) is equivalent to ∗∗ 1 v(u) λ′(φ(u)) = B v(q), 2 2 v(q) where B (T )= T 2 T + 1/6. We then set 2 − 1 v(u) λ′(φ(u)) := B v(q)+ v(θ(u)) 2 2 v(q) where (1 qmu)(1 qmu 1) θ(u) = (1 u) − − − , − (1 qm)2 mY1 ≥ − 34 BALAKRISHNAN,DAN-COHEN,KIM,ANDWEWERS and check that λ′ satisﬁes (i)–(iii)’. The proof of Chapter VI, Theorem 4.2 of Silverman [Sil2] applies with the real logarithm replaced by the p-adic logarithm throughout. This completes the proof of proposition 4.5.1.

4.6. Corollary. Suppose Fv is unramiﬁed over Ql and suppose Ev has semistable reduction. Let Nv = v(∆(Ev)). Then the possible values for φ on X( ) are v Ov (n(N n)/2N ) log l; 0 n < N . − v − v ≤ v Proof. By proposition 4.5.1, this follows from theorem 4.1.6. 5. Punctured elliptic curves of low rank 5.1. We put ourselves in the situation and the notation ( , , α, β, b, S, E X Nl, Wl, ...) of segment 1.12 with p an odd prime of good reduction, and T = S p , and with the goal of proving the theorem stated there, we begin by∪ computing { } the image of loc : Sel2( ) H1(G ,U ). p X → f p 2 We have the exact sequence 0 Q (1) U V (E) 0, → p → 2 → p → where Vp(E)= Tp(E) Qp is the Qp-Tate module of E = Q. We recall that ⊗ E⊗ 0 0 H (GT ,Vp(E)) = H (Gp,Vp(E)) = 0, so we have inclusions like so. H1(G , Q (1)) / H1(G , Q (1)) T _ p p _ p θ 1 1 H (GT ,U2) / H (Gp,U2) It is straightforward to check that in this context, maps of Galois modules send crystalline classes to crystalline classes, so these inclusions induce inclusions like so. 1 1 H (GT , Qp(1)) / H (Gp, Qp(1)) f _ f _

θf 1 / 1 Hf (GT ,U2) Hf (Gp,U2) ANON-ABELIANCONJECTURE 35

5.2. Lemma. If we assume, as in theorem 1.12, that (Z) has rank zero3 E and that Ш [p ] < , then Sel2( ) is contained in the image of θ . E ∞ ∞ X f Proof. It is a general fact (which is straightforward to check) that the map H1(G ,U ) H1(G ,U ) T n+1 → T n restricts to a map of Selmer schemes. On the other hand, we have an inclusion Sel1( ) Sel1( ), and X ⊂ E Sel1( )= (Q) Q = 0 E E ⊗ p by corollary 3.4.1, so Sel1( ) = 0. 2 X 1 Hence each P Sel ( ) is θ(Q) for some Q H (GT , Qp(1)). To see that Q is crystalline∈ at p, weX recall that ∈ H0(G ,V (E) B ) = 0, p p ⊗ cr which implies that the map θB in the following diagram H1(G , Q (1)) / H1(G , Q (1)) / H1(G ,B (1)) T _ p p _ p p _ cr

θ θB H1(G ,U ) / H1(G ,U ) / H1(G ,U B ) T 2 p 2 p 2 ⊗ cr is injective as shown, so that θ(Q) crystalline implies Q crystalline. 2 1 This allows us to regard Sel ( ) as a subset of Hf (GT , Qp(1)), and to 1 X compute its image in Hf (Gp, Qp(1)). 1 5.3. Recall (for instance from segment 6.2 of [DCW2]) that Hf (GT , Qp(1)) can be realized as the subspace of Q∗ Z Qp spanned by elements that are units outside S. Since Z Q = 0, we⊗ have ∗ ⊗ p 1 1 1 .((H (G , Q (1)) [Z ]∗ Z Q ֒ H (G , Q (1)) H (G , Q (1 f T p ≃ S ⊗ p → f p p ⊕ v p Mv S ∈ We define the function φ : H1(G , Q (1)) Q v v p → p by c log χ c 7→ ∪ (including v = p). We put these together to define φ : H1(G , Q (1)) H1(G , Q (1)) Q f p p ⊕ v p → p Mv S ∈ 3To avoid misunderstanding, we remind the reader that E refers to the compact curve, so that E(Z)= E(Q), where E is the generic fiber of E. That is, what we write as E(Z) is what is usually called the rational points of E, while our X (Z) is sometimes confusingly referred to as the integral points of E. 36 BALAKRISHNAN,DAN-COHEN,KIM,ANDWEWERS by φ((cv)v T )= φv(cv). ∈ Xv 5.4. Lemma. For elements c H1(G , Q (1)), we have ∈ f T p φ (locv c)v T = 0. ∈ Proof. We have

φ (locv c)v T = log χ locv(c) ∈ ∪ vXT ∈ = loc (log χ c) v ∪ vXT ∈ = loc (log χ c) v ∪ all placesX v since the contributions away from T vanish. By global class field theory (see, for instance, Tate [Tat, Theorem B, §11]), the composite H2(G , G ) H2(G , G ) Q/Z F m → v m → Mv is equal to zero. By Hilbert’s theorem 90, the cohomologies with µpn - coefficients inject into the cohomologies with Gm-coefficients. Taking inverse limits and tensoring with Qp, we find that the composite H2 G , Q (1) H2 G , Q (1) Q F p → v p → p Mv is equal to zero, which completes the proof of the lemma.

5.5. For l = p, we saw in corollary 4.6 that on jl (Zl) the function φv takes the values 6 X (n(N n)/2N ) log l, − l − l where Nl = ordl ∆ . As in the introduction, we deﬁne E n(N n) W := l − log l 0 n < N l 2N ≤ l l and W := Wl, Yl S ∈ and for w = (w ) W , we set l ∈ w := w . || || l Xl S ∈ ANON-ABELIANCONJECTURE 37

According to lemma 5.4, if c Sel2( ), we have ∈ X φ (loc (c)) = φ (loc (c)) = w p p − v v k k Xv S ∈ for some vector w W . ∈ Proposition. With assumptions as above we have loc (H1(U )) = η H1(G ,U 3 U 2) φ (η)= w . p Z 2 ∈ f p \ p k k w[W ∈ Proof. The inclusion has already been shown. To see that these equations deﬁne exactly the image,⊂ note that the local reciprocity law log χ k(a) = log χ(rec (a)) ∪ v for a Q shows that we get an isomorphism ∈ v∗ (*) (log χ) ( ) : H1(G , Q (1)) H1(G , Q (1)) ∪ · f p p ⊕ v p Mv S ∈ H2(G , Q (1)) H2(G , Q (1)) ≃ p p ⊕ v p Mv S ∈ Q . ≃ p Mv T ∈ 1 Indeed, for v = p, H (Gv, Qp(1)) is one dimensional and generated by the class of k(v). We6 see this by the exact sequence

0 Z∗ Q∗ Z 0 → v → v → → and the fact that the kernel has to map to zero under the Kummer map (since v = p). So it suffices to show that 6 log(χ(rec(v))) is non-zero in Qp. But χ(rec(v)) = χ(Frv) is just v Zp∗, an element of infinite order. So its log is non-zero. ∈ 1 For v = p, H (Gp, Qp(1)) is two-dimensional. But we’ve already discussed 1 the fact that Hf (Gp, Qp(1)) is one-dimensional, generated by the Kummer image of the units in Zp. Thus, it suffices to show that χ(rec(Zp∗)) is of infinite order. But in fact, χ(rec( )) just induces an isomorphism Z Aut (Z (1)) · p∗ ≃ p by the definition of the reciprocity map in local class field theory ([Ser, p. 146]). 1 The isomorphism (*) will take the (injective) image of Hf (GT , Qp(1)) to 2 the (injective) image of H (GT , Qp(1)), which is exactly equal to the kernel of the sum map Σ Q v Q . p −−→ p Mv T ∈ 38 BALAKRISHNAN,DAN-COHEN,KIM,ANDWEWERS

Since 1 dim H (G , Q (1)) = dim[(Z )∗ Q ]= T 1, f T p S ⊗ p | |− we see thereby that (log χ) ( ) takes H1(G , Q (1)) also isomorphically to ∪ · f T p the kernel of the sum map. On the other hand, we have seen that log χ : j ( (Z )) W Q . ∪ v X p ≃ v ⊂ p Mv S Mv S Mv S ∈ ∈ ∈ Therefore, the subspace H1(G, U ) H1(G , Q (1)), Z 2 ⊂ f T p which is defined as the inverse image of j ( (Z )), is exactly defined v S v X p by L ∈ η H1(G , Q (1)) φ (loc (η)) = w . ∈ f T p p p k k [w X 1 Hence, the p-component of elements of HZ(G, U2), that is, its image under locp, is exactly defined by the equations in the statement of the proposition.

5.6. Remark. According to proposition 5.4, the equality (Z)= (Z ) X X p 2 may be viewed as an exactness statement for the sequence h 1 (Z) (Z ) p Q , →X → X v −→ p vYT ∈ in a manner reminiscent of class ﬁeld theory. Since the map hp is quadratic, exactness here should be understood in the sense of pointed sets. Of course, this cannot hold literally, since we could take the image of an integral point in (Z ) and move it inside its residue disk just at one v = p without v T X v 6 changingQ ∈ the height. The formula (Z)= Ψ(w) (Z ), X ∪w ⊂X p ‘the projection to the p-component’, is one recasting of this exactness that absorbs the ambiguity. 5.7. We can now prove theorem 1.12. We follow the notation of [Kim4], section 3 and let Exp : LDR/F 0 H1(G ,U ) 2 ≃ f p 2 denote the non-abelian Bloch-Kato exponential map from the Lie algebra of DR the de Rham fundamental group. Recall that Ln may be realized as the DR st quotient of the tensor algebra T ·H1 ( Qp ) modulo the (n + 1) power of X DR the augmentation ideal. We denote by A, B the elements of L2 associated DR to the basis of H1 ( Qp ) dual to α, β . According to lemma 3.2 of [Kim4], DR 0 X { } L2 /F has basis A, A2, AB, BA . { } ANON-ABELIANCONJECTURE 39

According to the proof of corollary 0.2′ of [BKK], the map j : (Z ) H1(G ,U ) p X p → f p 1 is given by z jp(z) = log(z)Exp(A) = ( α)Exp(A), Zb while j : (Z ) H1(G ,U ) p X p → f p 2 is given by jp(z) = (log(z)Exp(A), D2(z) Exp([A, B])), where z D2(z)= αβ. Zb By Proposition 3.3 of [Kim4], we have

φp(Exp([A, B]) = 1. Therefore, φp(jp(z)) = D2(z). From this, we see that (Z ) = (Z )(tor) O. X p 1 E p \ For small primes p, it will happen frequently that (Z)= (Z ) , X X p 1 since global torsion on will often be equal to the local torsion for p small. But of course, this failsE for large p, and one must look at level 2, which then imposes on (Z ) the pair of conditions X p 2 log(z) = 0, D (z)= w 2 k k for some w, as in the statement of the theorem. 5.8. So far, we have tested conjecture 3.1 using the prime p = 5 for 256 semi-stable elliptic curves of rank zero from Cremona’s table, and found (Z)= (Z ) X X p 2 for each of them. To give a rough sense of the data computed using the methods of [Bal2], the details of which can be found on [Bal1], we present here a small table illustrating some of the large w -values that come up as we go through the list. k k 40 BALAKRISHNAN,DAN-COHEN,KIM,ANDWEWERS

Cremona number of Cremona number of label w -values label w -values || || || || 1122m1 128 3094d1 72 1122m2 384 3486o1 72 1122m4 84 3774f1 120 1254a2 140 4026g1 90 1302d1 72 4134b1 90 1302d2 96 4182h1 300 1426b4 64 4182h2 64 1506a2 112 4218b1 96 1806h1 120 4278j1 90 2397b1 72 4278j2 100 2418b2 64 4434c1 210 2442h1 78 4514d1 64 2442h2 84 4602b1 64 2478c2 68 4658d2 66 2706d2 120 4774e1 224 2967c1 72 4774e2 192 2982j1 160 4774e3 264 2982j2 140 4774e4 308 3054b1 108 4862d1 216

Hence, for example, for the curve 1122m2, y2 + xy = x3 41608x 90515392 − − there are 384 of the Ψ(w)’s that potentially make up (Zp)2. Of these, all but 4 end up being empty, while the points in those Ψ(Xw) consist exactly of the integral points (752, 17800), (752, 17048), (2864, 154024), (2864, 151160). − − 5.9. Another kind of test is to ﬁx a few curves and let p grow. For example, for the curve (‘378b3’) y2 + xy = x3 x2 1062x + 13590, − − we found that (Z ) = (Z)= (19, 9), (19, 10) X p 2 X { − − } for 5 p 97. As one might expect, as p gets large, (Zp)1 becomes signiﬁcantly≤ ≤ larger than (Z). For p = 97, we have X X (Z ) = 89. |X 97 1| However, imposing the additional constraint exactly cuts out the integral points for each p. ANON-ABELIANCONJECTURE 41

5.10. Is it conceivable that (Z) = Ψ (Z ) X tor w ⊂X p [w even when has higher rank? There are rather obvious relations with the conjecture onE non-degeneracy of the p-adic height [MST], which we hope to investigate in a later work. At the moment, we have a small bit of evidence, having tested the equality numerically for 10 curves of rank one, 2 curves of rank two, and one curve of rank three. Some of the cases are quite dramatic, such as Cremona label ‘82110bt2’, which has rank one. In this case, there are 2700 different w -values to consider, each contributing some Ψ(w). How- ever, the only non-emptyk k ones are those that contain the 14 integral torsion points [Bal1]. 5.11. We close this section with a brief mention of the framework for conjecture 3.1 when (Z) has rank one, leaving a systematic treatment to a later paper. As above,E denotes the regular minimal model of an elliptic curve over Q. Assume thatE there is a point y (Z) of infinite order. In the case where the Tamagawa number of is 1,∈X we saw in [Kim4, BKK] that E loc : H1(U ) H1(G ,U )= A2 p Z 2 → p p 2 is computed to be A1 A2; → t (t,ct2), 7→ where 2 c = D2(y)/ log (y). So the image is defined by x cx2 = 0. Meanwhile, 2 − 1 (Z ) H1(G ,U ) X p → f p 2 is z (log(z), D (z)). 7→ 2 Thus, (Z ) X p 2 is the zero set of D (z) c log2(z). 2 − It is sometimes convenient to write this defining equation as D (z) 2 = c. log2(z) In the earlier paper [BKK], we checked in a number of cases that the integral points do indeed fall into the zero set. However, it turns out that (Z) ( (Z ) X X p 2 42 BALAKRISHNAN,DAN-COHEN,KIM,ANDWEWERS in the majority of cases, underscoring the importance of going up another level. (In fact, a superficial guess based on the rank zero case would indicate that if the localization H1(G, U ) H1(G ,U ) Z n → f p n becomes injective at level n, then the conjecture might hold at level n + 2.) 5.12. When we assume that is semistable but with arbitrary Tamagawa numbers as above, the preciseE form of the equation becomes a bit delicate, and we will leave a systematic treatment to a later paper. However, if we define in this case h(y) c = , (log(y))2 where h is the p-adic height [MST] (except that our convention for the height function multiplies theirs by p) and y is a point of infinite order, then we can prove the following: Proposition. C (Z) z (Z ) D (z)+ (log(z))2 c(log(z))2 = w , X ⊂ { ∈X p | 2 2 − || ||} w[W ∈ where a2 + 4a E (E, α) C = 1 2 2 12 − 12 is the special value of the p-adic modular form E2 associated to the pair (E, α). The equations on the right hand side should in fact define (Zp)2, but we will not check this at the moment. X

Proof. Let hp(z) := hp(z O, z O) denote the local height at p of z. We ﬁrst show that − − C h (z)= D (z)+ (log(z))2. p 2 2 To do this, we use an interpretation of the local height at p in terms of Cole- man integrals as in Theorem 4.1 of [BB]. Note the following normalization: our global height (and local heights) are precisely half those in [BB]. In terms of our normalization of local heights, Theorem 4.1 of [BB] gives that z hp(z O, z O)= αη, − − − Zb with [η] [α] = 1. Note that η, which is found in the course of proving Corollary∪ 4.2 of [BB], is given by η = η + Cα, − 0 where η0 = β and a2 + 4a E (E, α) C = 1 2 2 12 − 12 ANON-ABELIANCONJECTURE 43 is the special value of the p-adic modular form E2 associated to the pair (E, α). Then substituting appropriately, we have z hp(z)= αη − Zb z z = αη0 + C αα Zb Zb C = D (z)+ (log(z))2. 2 2 Thus we have that C h (z)= D (z)+ (log(z))2. p 2 2 Finally, since h(y) h(z) c = = , (log(y))2 (log(z))2 we have c(log(z))2 = h(z) C = D (z)+ (log(z))2 + h (z O, z O), 2 2 v − − Xv=p 6 and noting that the possible values of h (z O, z O) on integral points − v − − z are precisely given by φv, we conclude that

C (Z) z (Z ) D (z)+ (log(z))2 c(log(z))2 = w . X ⊂ { ∈X p | 2 2 − || ||} w[W ∈

6. The thrice punctured line 6.1. Let = P1 0, 1, and take b to be the standard tangential base- X \ { ∞} point −→01 based at 0 P1. For the basic facts here, we refer to [Kim2]. We have ∈ U = Q (1) Q (1) 1 p × p and 2 3 U /U = Qp(2), so there is an exact sequence 0 Q (2) U Q (1) Q (1) 0. → p → 2 → p × p → The diagram (Z) / (Z ) X X p

loc 1 p / 1 HZ(G, U1) Hf (Gp,U1) 44 BALAKRISHNAN,DAN-COHEN,KIM,ANDWEWERS thus becomes / (Z ) ∅ X p

locp 0 / A2. The map jp takes the form z (log(z), log(1 z)), 7→ − (see, for instance, Proposition 7.3 of Dan-Cohen–Wewers [DCW2]) so that (Z ) is the common zero set of log(z) and log(1 z). Since z and 1 z X p 1 − − must both be roots of unity, the only common zero possible is z = ζ6 for a primitive sixth root of unity. If p = 3 or p 2 mod3, then ζ6 / Qp, so that (Z ) = φ. That is, ≡ ∈ X p 1 Proposition. Conjecture 3.1 is true for n = 1 when = P1 0, 1, and p = 3 or p 2 mod3. X \ { ∞} ≡ 6.2. When p 1 mod3 ≡ (Z)= φ ( ζ ,ζ5 = (Z ) X { 6 6 } X p 1 and we must go to a higher level. We have 1 3 Hf (Gp,U2)= A and j : (Z ) A3 p X p → is given by j (z) = (log(z), log(1 z), Li (z)), p − − 2 where zn z Li2(z)= = (dt/t)(dt/(1 t)) n2 Z − Xn b is the p-adic dilogarithm (loc. cit.). Meanwhile, we still have Sel1( ) = 0, X 1 since H (GT , Qp(2)) = 0 by Soulé’s vanishing theorem [Sou]. Therefore, (Z ) = z log(z) = 0, log(1 z) = 0, Li (z) = 0 , X p 2 { | − 2 } and the question of whether (Z) = (Zp)2 reduces to checking if Li2(ζ6) can be zero. Note that ([Col],X Prop. 6.4)X 1 2 Li (ζ )+ Li (ζ− )= log (ζ )/2 = 0, 2 6 2 6 − 6 so that we need only discuss non-vanishing at one of the sixth roots. This question was raised by Coleman in [Col], page 207, remark 3. ANON-ABELIANCONJECTURE 45

6.3. We have checked numerically thus far that Li (ζ ) = 0 2 6 6 for p in the range 3 p< 105. This may be carried out as follows. We define power series g Q≤[[v]] recursively by n ∈ (v 1)p g (v)= v 1 − 0 − − vp (v 1)p − − and for n 1, ≥ 1 2 g′ (v)= v− g (v)(1 + v + v + ) n+1 − n · · · gn+1(0) = 0. Then p2 1 Li (ζ)= g ( ) 2 p2 1 2 1 ζ − − for any (p 1)st root of unity ζ. Indeed, this is a special case of Propositions − 4.2 and 4.3 of [BdJ]. Hence, since 1/(1 ζ6) = ζ6, it suffices to check that g (ζ ) = 0. In fact, (for p as above) g (ζ−) is nonzero modulo p. Moreover, 2 6 6 2 6 g2 reduces modulo p to a polynomial of degree p 2 which is determined by the reductions modulo p of the same equations as− above. So the verification may be performed rapidly with any computational software. We used Sage [DCW1]. 6.4. However, this may fall short of providing definitive evidence that Li (ζ ) = 0 2 6 6 for all primes congruent to 1 mod 3: if for each p, the value of g2(ζ6) modulo p were merely random, the probability of g2(ζ6) being nonzero for p in our range would be about 0.413. On the other hand, the probability that g(ζ6) = 0 for some p 1 mod3 would be 1. The point is that the product ≡ (1 1/p) − p=1Y mod3 tends to zero, representing the probability that g2(ζ6) does not vanish mod p for all p 1 mod3 (assuming these values are random, independently and evenly distributed≡ variables). Suppose we want to falsify the randomness hypothesis. To do this, we could show that g2(ζ6) doesn’t vanish for all p < N for some large value of N. Unfortunately, the convergence of the product is extremely slow: for N = 100, 000 it is just 0.413, which does not give convincing evidence. This computation took several hours. Doing the computation up to p< 106 would take several days, and the probability would be 0.3775, which is not so much better. 46 BALAKRISHNAN,DAN-COHEN,KIM,ANDWEWERS

7. Remarks on curves of higher genus 7.1. Let Spec Z be the regular minimal model of a proper smooth curve of genusX → 2 (case 2 of the trichotomy of segment 2.1). We ﬁx a base point b (Z)≥. Then the associated map ∈X j : (Z ) H1(G ,U ) p X p → f p 1 can be identiﬁed with the map , Z ) X(Q ) ֒ J (Q ) T J) X p ⊂ p → X p → e X where JX is the Jacobian of X and TeJX is its tangent space at the origin [BK]. 7.2. If we assume that Ш [p ] < , then it follows that JX ∞ ∞ J (Z) Q = Sel1( ). X ⊗ p X Indeed, the corollary of segment 3.4.1 applies equally to the abelian variety JX : J (Z) Q = Sel1(J ). X ⊗ p X Since the map U1( ) U1(JX ) is an isomorphism, we have a natural inclusion X → Sel1( ) Sel1(J ). X ⊂ X Conversely, if P Sel1(J ) ∈ X is an arbitrary Selmer class for the Jacobian, then loc P = 0 for all v = p v 6 whence P Sel1( ). ∈ X 7.3. If we assume moreover JX (Z) has rank zero, then it follows from segment 7.2 that Sel1( ) = 0. So by segment 7.1 we have X (Z ) = (Z ) J (Z )(tor). X p 1 X p ∩ X p 7.4. We apply this to the Fermat curve l l l Xl : x + y = z for prime l 5. It is a theorem of Coleman, Tamagawa, and Tzermias [CTT, Theorem 2]≥ that X (Z ) J (Z )(tor) l p ∩ Xl p must satisfy xyz = 0. Therefore, if ζ / Q , then l ∈ p X (Z)= X (Z ) J (Z )(tor). l l p ∩ Xl p (Using also the theorem of Wiles.) Meanwhile, we have

rank JXl (Z) = 0 for l = 5, 7 [Fad]. So by segment 7.3, we have ANON-ABELIANCONJECTURE 47

Proposition. With notation as above, assume

ШJ [p∞] < . Xl ∞ If is the minimal regular model of X , we have Xl l (Z)= (Z ) Xl Xl p 1 for l = 5, 7 and p 1 mod l. That is, conjecture 3.1 is true at level 1 for these l and p. 6≡ 7.5. It should be interesting to investigate conjecture 3.1 in relation to the many known results about torsion packets on curves, for example, for Fermat curves or modular curves [BR]. It seems reasonable to suspect that there should be more instances where (Z) = (Zp)1 even when the Jacobian has positive rank. Another wayX to say thisX is that the classical method of Chabauty is usually applied with one choice of a differential form. The question here raised by conjecture 3.1 is how often the common zero set of all available abelian integrals will give us exactly (Z). X 7.6. In fact, it is relatively easy to produce example of affine curves of higher genus and good reduction primes p for which the conjecture holds. We illustrate this by way of an example. Consider the elliptic curve E with affine model y2 = x3 891x + 4374. − It turns out that E(Q) Z/4 ≃ and that the two points P and Q of order four are ( 9, 108). Thus, by making the substitution x +9= dt2, we get a cover − ± f : X E O → \ { } ramified exactly over P and Q. This affine hyperelliptic curve has equation y2 = d3x6 27d2x4 648dx2 + 11664. − − For any p of good reduction and d such that d is not a square in Qp, the point of order two (27, 0) will not lift to (Zp). On the other hand, we have the commutative diagrams X (Z) / (Z ) X X p f ∼= [ O](Z) / [ O](Z ) E \ E \ p (Z ) / H1(G ,πQp (X,¯ b)) X p f p 1

[ O](Z ) / H1(G ,πQp (E¯ O, b)) E \ p f p 1 \ 48 BALAKRISHNAN,DAN-COHEN,KIM,ANDWEWERS and 1 Qp ¯ / 1 Qp ¯ HZ(G, π1 (X, b)) Hf (Gp,π1 (X, b))

H1(G, πQp (E¯ O, b)) / H1(G ,πQp (E¯ O, b)). Z 1 \ f p 1 \ These imply that 1 (Z ) f − [ O](Z ) . X p 2 ⊂ E \ p 2 Hence, for such d and p, it is easy to deduce that whenever [ O](Z) = [ O](Z ) , E \ E \ p 2 we also get (Z)= (Z ) X X p 2 for free. We can check this, for example, for d = 2 and all 3 p 53 such that p 3 mod8 and p 5 mod8. ≤ ≤ ≡ ≡ 8. Remarks on S-integral points 8.1. Let Spec Z denote a regular Z-model of a hyperbolic curve over Q, and letXb →denote a base point as in segment 2.1. As above we denote by Un the level-n quotient of the unipotent p-adic étale fundamental group of ¯ at b. We also use the same notation for the fundamental group of ¯ . XQ XQp Let S denote a finite set of primes of Z, p a prime of good reduction not in S, and let T be a finite set of primes containing S and p, as well as all primes of bad reduction. We define the S-integral Selmer scheme of in level n by X n 1 1 Sel ( )= loc− [Im(j )] H (G ,U ). S X v v ⊂ f T n v=\p,v/S 6 ∈ This gives rise to a filtration 1 1 (Z ) := j− (loc (H (U ))) X p S,n p p Z,S n on (Z ), which one might conjecture to converge to (Z ). X p X S 8.2. As an (admittedly small) step in this direction, we consider the case = P1 0, 1, and S = 2 . Then the diagram X \ { ∞} { } (Z[1/2]) / (Z ) X X p

j jp 2 / 1 SelS( ) Hf (Gp,U2) X locp ANON-ABELIANCONJECTURE 49 becomes 2, 1/2, 1 / (Z ) { − } X p j jp A2 / A3 locp where [DCW2] 2 locp(x,y) = ((log 2)x, (log 2)y, (1/2)(log 2)xy). Recall that j (z) = (log(z), log(1 z), Li (z)), p − − 2 so that (Z ) is the zero set of X p 2 2Li (z) + log(z) log(1 z). 2 − The fact that 2, 1, 1/2 is in the zero set, which we deduce here from the commutativity of{ the− localization} diagram for Selmer schemes, was noticed earlier by Coleman to be a consequence of standard dilogarithm identities ([Col], remark on page 198). We have checked numerically that this is exactly the zero set for p = 3, 5, 7, so that

(Z[1/2]) = (Zp) 2 ,2 X X { } in that case. The equality starts failing for larger p. Coleman already noted 1 √5 this failure for p = 11, since − ±2 , for example, is in the zero set. This fact, as well as the considerations of the previous sections, indicate the importance of investigating systematically weakly global points of higher level. References

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J.B.: Boston University, Department of Mathematics and Statistics, 111 Cummington Mall, Boston MA 02215, USA M.K: Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG, and Department of Mathematics, Ewha Women’s University, 52 Ewha-yeo-dae-gil, Seoul, Korea 120-750 I.D.: Department of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva, Israel S.W.: Institut für Reine Mathematik, Universität Ulm, Helmholtzstrasse 18, 89081 Ulm, Germany